A088322 Number of monotone functions f: 2^X -> 2^X where 2^X is the power set of an n-set X. Here f is monotone means that if A is a subset of B then f(A) is a subset of f(B).

1, 3, 36, 8000, 796594176, 25039893834551321901, 230156231509903526722108570920314496786496, 478651764962008689839230538296564128023598629748415103570025502338085999191479922367872

Offset: 0

Author

W. Edwin Clark, Nov 06 2003

Keywords

• nonn

Comments

Proof of formula by Robert Israel: If f is monotone, then for each x in X the set G(x) = {A in 2^X: x in f(A)} is an upset, i.e. if A is in G(x) and A \subset B then B is in G(x). Conversely, if for each x in X the set G(x) is an upset, then f is monotone. And the family {G(x): x in X} determines f, since f(A) = {x: A is in G(x)}. So the cardinality of the set of monotone set-functions is N(|X|)^|X| where N(|X|) is the cardinality of the set of upsets G of 2^X, or equivalently monotone Boolean functions. That is sequence A000372.

This sequence was motivated by a question by Federico Echenique on sci.math.research.

Crossrefs

Cf. A000372, A061301.

Formula

a(n) = A000372(n)^n.